The National Sleep Foundation used a survey to determine whe

The National Sleep Foundation used a survey to determine whether hours of sleep per night are independent of age (Newsweek, January 19, 2004). A sample of individuals was asked to indicate the number of hours of sleep per night with categorical options: fewer than 6 hours, 6 to 6.9 hours, 7 to 7.9 hours and 8 hours or more. Later in the survey, the individuals were asked to indicate their age with categorical options: age 39 or younger and age 40 or older. Sample data follow.

Conduct a test of independence to determine whether hours of sleep are independent of age. Using a .05 level of significance.

Compute the value of the ² test statistic (to 2 decimals):

The p-value is:

.

What is your conclusion?

b. What is your estimate of the percentages of individuals who sleep fewer than 6 hours, 6 to 6.9 hours, 7 to 7.9 hours and 8 hours or more per night? Round your answer to one decimal place.

Hours Of Sleep	39 or Younger	40 or Older
Fewer than 6	38	36
6 to 6.9	60	57
7 to 7.9	77	75
8 or More	65	92

Solution

The National Sleep Foundation used a survey to determine whether hours of sleep per night are independent of age (Newsweek, January 19, 2004). A sample of individuals was asked to indicate the number of hours of sleep per night with categorical options: fewer than 6 hours, 6 to 6.9 hours, 7 to 7.9 hours and 8 hours or more. Later in the survey, the individuals were asked to indicate their age with categorical options: age 39 or younger and age 40 or older. Sample data follow.

Hours Of Sleep

39 or Younger

40 or Older

Fewer than 6

38

36

6 to 6.9

60

57

7 to 7.9

77

75

8 or More

65

92

Conduct a test of independence to determine whether hours of sleep are independent of age. Using a .05 level of significance.

Chi-Square Test

Observed Frequencies

Column variable

Calculations

Hours Of Sleep

39 or Younger

40 or Older

Total

fo-fe

Fewer than 6

38

36

74

2.48

-2.48

6 to 6.9

60

57

117

3.84

-3.84

7 to 7.9

77

75

152

4.04

-4.04

8 or More

65

92

157

-10.36

10.36

Total

240

260

500

Expected Frequencies

Column variable

Hours Of Sleep

39 or Younger

40 or Older

Total

(fo-fe)^2/fe

Fewer than 6

35.52

38.48

74

0.1732

0.1598

6 to 6.9

56.16

60.84

117

0.2626

0.2424

7 to 7.9

72.96

79.04

152

0.2237

0.2065

8 or More

75.36

81.64

157

1.4242

1.3147

Total

240

260

500

Data

Level of Significance

0.05

Number of Rows

4

Number of Columns

2

Degrees of Freedom

3

Results

Critical Value

7.8147

Chi-Square Test Statistic

4.0070

p-Value

0.2607

Compute the value of the ² test statistic (to 2 decimals): 4.01

The p-value is:   0.2607

.

What is your conclusion?

Calculated P=0.2607 > 0.05 level of significance.

The null hypothesis is not rejected.

We conclude that hours of sleep are independent of age.

b. What is your estimate of the percentages of individuals who sleep fewer than 6 hours, 6 to 6.9 hours, 7 to 7.9 hours and 8 hours or more per night? Round your answer to one decimal place.

Fewer than 6

14.8%

6 to 6.9

23.4%

7 to 7.9

30.4%

8 or more

31.4%

Hours Of Sleep	39 or Younger	40 or Older
Fewer than 6	38	36
6 to 6.9	60	57
7 to 7.9	77	75
8 or More	65	92